How can never ending decimal numbers represent finite lengths? e.g. $\pi$, $\sqrt{2}$

Question

Recently, I was in a discussion with a colleague that, whether the $\pi d$ really can represent the accurate perimeter of a circle or not. To clarify that doubt, I came here and found below similar post:
a Circle perimeter as expression of ππ Conflict?

Even though above Q is quite similar, the answer is not satisfactory for my doubt. So I decided to represent it in different way.

Now a perfect circle is not possible in real world due to various other reasons. However for sake of argument, let's assume that we have a reasonably thin string of a finite length $x$ (say $10.\bar{0}$ cm). Using that string, we made a circle of diameter $d$.

Now here is a paradox:

1. $x$ is measurable using simple foot ruler, hence finite;
2. The theoretical perimeter of the circle is $\pi d$;

How can we equate?

$$x = \pi d$$

If I assign a task of $x$ to be measured, then it will be measured perfectly using a simple foot ruler to exact $10$ cm. But $\pi d$ can't be measured perfectly even using super computer as the value of $\pi$ goes on and on for trillions of decimals.

Does $\pi d$ represents the perfect perimeter of a circle or the nearest value?

Same can be asked for a square which has sides of $1$ unit and diagonal of $\sqrt{2}$. Here, how the finite length is measured using a never ending decimal?

Answer 1

You set up a false dilemma. How many digits a decimal representation of a number has only tells us how much information is needed in the decimal system to describe the number.

While reading the decimal digits of $\pi$ we gain more and more detail about the exact value of the number:

$$ \begin{align} \pi &= 3.1...&\implies&&3.1\leq&\pi\leq3.2\\ \pi&=3.14...&\implies&&3.14\leq&\pi\leq3.15\\ \vdots&&\vdots&&&\vdots\\ \pi&=3.141592...&\implies&&3.141592\leq&\pi\leq3.141593\\ \end{align} $$ so that there are infinitely many digits only goes to show that our decimal system is not "powerful" enough to give all details about the number $\pi$ as a finite set of data.

The size of the data describing $\pi$ in a given system of representation bears no witness to the size of the number itself.

---

An experiment to consider

You may have the idea that you can measure any given distance with perfect precision, but try the following experiment:

Draw a straight line of random length on a piece of paper. Then measure it using a ruler - chances are that it will not fit exactly from mark to mark.

Suppose then from a theoretical point of view that we had a decimal system ruler with infinitely fine markings on it. Then you could zoom in to the $3.14$ and $3.15$ marks and recognize that $\pi$ lies somewhere between those two - much closer to the $3.14$ mark than to the other.

After that try zooming in quite a deal more to the $3.141592$ and the $3.141593$ marks and again $\pi$ escapes fitting any of those two exactly. It is impossible to perform this experiment in practice, but actually the same phenomenon is most likely to be the case for your randomly drawn line - if you only had the power to keep zooming in.

Answer 2

Two different and independent long essay answers:

Essay 1:

To elaborate on the many comments and answers I think there few fundamental concepts that are troubling.

The first is what I'd call basic assumption that there must be a "quantum atom" some basic smallest measure where everything meshes harmonically. If something doesn't measure in our unit of measurement exactly: then we divide divide our basic unit of measurements into a discreet number of smaller units, if doesn't mesh then we keep dividing, eventually we will find a precise unit it does mesh up to. Pythogoras assumed this was true. The $\sqrt{2}$ shows this isn't true. To be fair to Pythogoras, he didn't think the quantum atom atom had to be power of 10, or 2 or 19 or any individual number but he did think that for any measurable value there'd be a harmonic ratio of whole numbers, an $a$ and a $b$ so that the value would be $\frac a b$. He was wrong.

In physics, or what the OP called "the real world", the idea of atoms are the smallest unit from which you build up. In the "real world" that seems to be the quantum level and about something like $10^{-23}$ meters (I'm talking off the top of my head here; I do not know anything about physics other than popular literature) where the ability for space and distance to even exist breaks down (so far as I understand it-- which is very little). If math has "atoms", they are actually very large. They are the whole numbers from which we build up and from which we build down. By taken smaller and smaller powers of ten, we aren't going to reach bedrock. 0 and 1 was the bedrock. Miniscule powers of 10 is just sand.

Second is the "how many angels can dance on the head of pin" issue. 3 < 3 + .1 = 3.1 < 3.1 + .04 = 3.14 < 3.14 + .001 = .... We have an infinite sum that we keep adding things onto so it keeps getting bigger. If something gets bigger and bigger and never stops getting bigger it must be unlimited and therefore infinite, it would seem.

BUT Notice if we took an extra step in the infinite sum. $3 < \pi$. Let $\epsilon_1 = \pi - 3$. $.1 < \epsilon_1 < .2$. So $3.1 < \pi$. Let $\epsilon_2 = \pi - 3.1$. $.04 < \epsilon_2 < .05$, etc. Although we are infinitely adding things, the things we are adding are always less than enough to get to a precise finite amount. The sum may be infinite in mechanics but it is not infinite in value because each step of addition has the same distinct upper bound.

If you don't like the unexpressable value $\pi$, we can do the same thing for the very basic unit $1$. Start with $1/2$. Then add $1/4$. Then $1/8$. Each step of the way we end up less than $1$. So then we add something but we specifically add something less than where we want to get to. That way we will have an infinite sum. $1/2 < 1$; $1 - 1/2 = 1/2$ so we choose $1/4 < 1/2$. $1/2 + 1/4 = 3/4 < 1$. $1 = 3/4 = 1/4$ so we choose $1/8 < 1/4 $ so $1/2 + 1/4 + 1/8 = 78 < 1$. We can do this forever and always be bounded above by 1.

This dovetails into the idea that we can find things infinitely close to 0. If $d > 0$ then we can always find $d > d/2 > d/4 > d/8 > ....$ all greater than 0. This means we can take an infinite sum that is always less than a finite $x$. $x - 1/2 < x - 1/4 < x - 1/8 < ....$. From there it's straightforward to realize $3 < 3.1 < 3.14 < 3.141 < 3.1415 < ..... < \pi < .... <3.1415 < 3.142 < 3.15 < 3.2 < 4$.

Thirdly, there is the idea that elementary school children of the 20th and 21th have been taught from the very beginning that we can use decimals to express everything. This isn't actually true. We can't even use decimals to express $1/3$. But we learn about handwaving and "rounding errors" and know that $0.33333....$ has an infinite number of 3s and they represent smaller values that eventually we can ignore them and... oh, it'll work out somehow.

The thing is we can't express everything as decimals. But with real analysis we know that the real numbers have the least upper bound property and that the rationals are dense in the reals. So for any value, $\pi$, $\sqrt{2}$, $1/3$ we can find an infinite sequence of rational numbers whose limit tends to the value. Knowing that we can realize that decimals are a perfect tool to create these sequences. We can make a {3, 3.1, 3.14, 3.141, 3.1415,...} sequence that converges to $\pi$; we can make a {0.3, 0.33, 0.333, ....} sequence that converges to $1/3$.

Because the reals have the least upper bound property, every real value is the limmit of an infinite decimal and every infinite decimal converges to a real value. This is big. And we teach it to our children. But we misunderstand it and take it a message given by God on High that decimal numbers are everything and all there is. And then when we learn that the decimal expression of $\pi$ never ends... we freak out. It jut doesn't make sense to us. Everything must be expressable as a decimal and if we don't know what the decimal is then it can't really have a value, can it?

Well, no. As Pythagoras painfully discovered 3,000 years ago, everything doesn't have a rational (which is equivalent to decimal) expression. What we actually discovered 200 years ago is everything is the limit of an infinite decimal expansion. This is subtlely but crucially different.

Then the 4th thing: If we can't express values as decimals then how can we express them. This is very hard to wrap our heads about, but the simple answer is: we can't. There are irrational numbers such as $\sqrt{2}$ and $\sqrt[8]{7 + \sqrt[15]9}$ which we can express as solutions to equations, and there are specific useful numbers like $\pi$ or $e$ that have useful values. But there are uncountably more irrational numbers that we simply can not have any means of describing or expressing. Imagine an infinite decimal where each value is an arbitrary value 0-9 determined with no pattern or "meaning". We simply can not express or describe that number in any meaningful way. .... Oh, well.

======

2nd long essay answer.

You are imagining this as being given from Mathematicians On High that $\pi$ has the infinite expansion $3.141592653....$ and trying to work backwards what does $3.141592653....$ mean really. Instead imagine going the other way. You have been told by Mathematicians On High (archimedes, actually) that all circles have the same ratio of circumference to diameter. Imagine you try to go forward to figure out what that value is. And suppose you don't have decimals yet. Just fractions.

"A: So, I'll take this rope ruler and I see that I can fit more that 3 diameters. It looks like it's about 3 and 1/7. Let's look closer... shoot it's a tiny bit less than 3 1/7. Let me divide that 1/7 bit into smaller pieces. Wow if I divide that into 120ths I get that $\pi = 3 + 1/7 - 1/120*7$ and then and then if I divide that into 15ths I get $\pi = 3 + 1/7 - 1/120*7 + 1/15*120*7$. Are you getting all this, B..."

B: "May I make a suggestion, A? Instead of taking arbitrary fractions, it'd be easier for me to take notes if we measured them all be a single ratio. Say 4, or better yet 10 as I have 10 fingers."

A:" Okay, 3 and then between 1/10 and 2/10s. So $\pi = 3 + 1/10$ and going further we get $\pi = 3 + 1/10 + 4/100 + 1/1000 + ...$ But wait. What if it turns out $\pi$ isn't an even sum of powers of 10. What if pi is something like $97/31$ and never resolves to a power of 10?"

B: takes out a calculator "Hmm... Oh! that'd be fine. It starts repeating in the 15th decimal place. $97/31 = 3 + 129032258064516/1000000000000000 + 4/31*1000000000000000$"

A: "Okay. I guess...I mean, if you say so. So, $3 + 1/10 + 4/100 + 1/1000 + 5/10,000 + 9/100,000 + .."

B: "Oh, wait. I just thought of something. What if there isn't any common denominator? what if $\pi$ is irrational? Like that $\sqrt{2}$ thing."

A: "Hmm, then I guess we'll be calculating forever. That doesn't seem right does it?"

B: "I don't know. I guess all that would mean is that we'd never reach a point the marks will match exactly. At every point, we divide the remaining amount be 10. There's no reason to assume that eventually it has land on exactly one of those 1/10 marks. So I guess there's no reason in can't go on forever. We aren't making anything bigger, after all. We're just making things finer. Smaller actually, and there's no reason things can't get infinitely precise. That's a different story than things getting infinitely big.

"Well, let continue."

A: "3 + 1/10 + 4/100 + 1/1,000 + 5/10,000 + 9/100,000 + 2/1,000,000 + 6/10,000,000 + ..."

B: "Hey, how are you figuring out these values anyway? Do you have a microscope and an overlapping ruler scale or something. Six ten-millionths is surprisingly precise. How'd you get that?"

A: "I don't know. That wasn't a criterion for this dialog."

B: "Oh, well"

B: "+ 5/100,000,000 + 8/1,000,000,000 + 5 + 5/10,000,000,000 + ..."

===== long essay answer 3 =====

We agree that pi is not infinite.

So how can 3 < 3.1 < 3.14 < ..... which is infinite and increasing not be infinite.

And the answer to that is that each inequality is bounded:

$3 < \pi < 4$

$3.1 < \pi < 3.2$

$3.14 < \pi < 3.15$

so we don't have a sequence that is increasing unlimitedly. We have a sequence which, although it is infinite and it is increasing, it is bounded in it's maximum possible value.

Such a sequence is called a Cauchy Sequence. I won't give the formal definition of a cauchy sequence but informally it is this: It is an infinite sequence of numbers such that at some point all the rest of the numbers will be within a small finite distance from each other and at another further point all the rest of the numbers will be withing an even smaller distance of each other and for any distance no matter how small, there will be a point in the sequence that all the remaining terms will be within that distance.

For example. {3, 3.1, 3.14, 3.141, 3.1415....} and {1/2, 3/4, 5/8, 11/16, 21/32, 43/64...} are cauchy sequences. In the first one all the terms are withing 1 of each other. After the first term they are all within .1 of each other. After the third term they are all withing .01 of each other. If I wanted to find a point where they were all within a billionth of each other I could. A googolth? I could do that too.

The second sequence is also a cauchy sequence because all the terms are within 1/2 of each other. After n terms all the remaining terms are all within $(1/2)^n$ of each other.

Now think about that for a moment. I don't want to give a formal analysis proof because they are abstract and hard to follow but think of what this means. At a certain point, all the remaining terms are "bounded" withing a tiny distance to each other. So even though there are an infinite number of them and even though it's possible they might all be increasing, all the terms are bounded within a certain distance so there is a very finite limit for all these terms to fall within. They can not ever get bigger than that. And is the bounding difference between the terms gets smaller and can be arbitrarily small all these terms telescope and bind themselves to a certain number that is the limit of all the terms.

Reread that last paragraph a few times. What I mean to say is that these infinite numbers all get infinitely close together and there is one precise value that the hone into. We call that number the limit and it exists for all cauchy sequences even though the sequence is infinite.

This is a fundamental theorem of analysis is that in the real numbers every number is a limit of cauchy sequences and every cauchy sequence converges to a number.

This is why decimal numbers are possible. We can't express 1/7 or $\sqrt{2}$ or $\pi$ in decimals. But we know we can find a series of rational numbers that get infinitesimally close to each other that hone into these numbers. Decimals are a perfect and natural way to do this. Each decimal point hones the number in ten time closer than the previous number. So a sequence of increasingly long, increasingly precise decimal numbers form a cauchy sequence that has a limit to a real number.

That is why it is okay to say that every real number is expressible as a (maybe infinite) decimal and why every decimal (even if it is infinite) represents a real number.

And if the number is irrational, it will need an infinite decimal.

Answer 3

As @Joanpemo pointed out you can't make a perfect circle so in practice it would not find the exact value of $\pi$. As you mentioned in the comments "How can a never ending decimal represent a finite length.", I believe to tackle this question it's important to consider Zeno's paradox of the tortoise and achilles (refer to Numberphile and this Ted-Ed Animation).

In Zeno's Paradox it seems as if Achilles will never reach the tortoise, but something Zeno hadn't touched on yet was the concept of a limit. We say that the limit is the value that the "function" or "sequence" approaches as the input approaches some value. In Zeno's Paradox it's the limit of the sequence of numbers, it's limit is 2. Yet it has an infinite sequence of numbers,$\frac {1}{2}, \frac {1}{4}, \frac {1}{8}, \frac {1}{16},...$ but it's limit is 2 (a finite number).

In the comments @Mathematician42 states how the decimal value $0.9999...$, which is an infinitely long decimal value is equal to 1 (a finite number). This can be shown by the following:

$$x = 0.999...$$ $$10x = 9.999...$$ $$9x = 9$$ $$\therefore x = 1$$

But with regards to $\pi$ I am by no means stating that it can be represented by a finite number, nor is the sequence of Zeno. But as it tends off into infinity it won't get any bigger than a limit. So just like in Zeno's paradox it gets infinitely small and never gets any bigger.

I hope this doesn't confuse you too much.

Answer 4

Setting physics and measurement procedures aside, what you have is that with a 10 cm string the measure of the diameter will not be a decimal number with a finite number of digits.

In fact when it comes to decimal representation, $\pi$ having infinite non-repeating digits is not special at all: almost all real numbers behave that way except for a few really extraordinary cases that are the rational numbers.

There are also several other points in your paradox that can be resolved; two examples:

• $\pi d$ is the perfect length of the string. You cannot measure exactly either $\pi$ or $d$ but they still exist. It's just that the product of two "ugly" numbers happens to be a "nice" one.
• you cannot measure anything perfectly. There is only one thing in the universe for which you can say that you perfectly know its length and that thing is the single measuring stick that you picked as your unit of measure. Any other length you measure will be an approximation with some margin of uncertainty. This means you cannot measure perfectly both the string and the diameter.

Answer 5

You are probably thinking with the implicit hypothesis that "measuring" a decimal takes a finite amount of time, so that an infinite number of decimals takes infinite time and isn't achievable.

In such thought-experiments, you should assume instead that the measure takes no time, or a time that goes decreasing with every decimal, sufficiently rapidly that the sum is finite. Then the paradox disappears.

Another approach is by saying that you can get an approximation as accurate as you want by increasing the number of decimals. There is no principled limit to the process.

Answer 6

Measuring of $\pi$

Nothing can be measured perfectly in the real world. Let us imagine that we have a device that measures length up to precision of $\varepsilon>0$ and try to measure length denoted by $\pi$.$^{[1]}$ What we get is some $x$ finitely represented with respect to some measuring unit (usually, we will get $x=3.1415\ldots$ if we choose decimal system). In this way, we do not know that $\pi = x$, but only that $\pi\in\langle x - \varepsilon, x+\varepsilon\rangle$. Smaller the $\varepsilon$, more precisely we know $\pi$.

What we conclude is that we cannot ever know exact value of $\pi$ by measuring it, even if $\pi$ had finite decimal representation.$^{[2]}$ Of course, if $\pi$ were rational$^{[3]}$, we would just calculate it's exact value and that would be the end of discussion. Since this is not the case, if we want to know about $\pi$, we need to have a mathematical device that would treat the problem. Such devices are called convergent sequences.

How does this work? Imagine that we construct a sequence $(a_n)$ and prove abstractly that it converges to $\pi$. Now we do know that for any precision $\varepsilon > 0$ there exists $n_0\in\mathbb N$ such that $|a_n-\pi|<\varepsilon$ for all $n\geq n_0$. Note that effectively finding what $n_0$ gives desired precision is entirely different matter and belongs to field of numerical analysis. This wiki article will give much more insight in how $\pi$ is actually computed.

Infinite representations giving finite length

In frog1944's answer there is beautiful mention of Zeno's paradox, which is how probably these kind of discussions formally began in western philosophy, as far as I know.

Imagine that you need to cross a room, but are doing it in following steps: firstly, you cross half the distance, then you cross half the remaining distance, and repeat the process infinitely. Will you reach your goal? Or will you cross infinite distance?

Mathematically, what we need to calculate is infinite sum of the form $$s = \frac 12+\frac 14 + \frac 18+\ldots$$ which is just the sum of geometric series $$\sum_{n=1}^\infty \frac 1{2^n} = 1$$ If you draw this process on paper, it will immediately be clear why such equality holds. And it also solves our problem of crossing a room: we will cross the room (exactly once), but not in finite number of steps.

Decimal representations work in exactly same manner; any number $x\in\langle 0,1\rangle$ can be represented as infinite sum of the form $$x = \frac{a_1}{10}+ \frac{a_2}{100}+\frac{a_3}{1000}+\ldots$$ where $a_1,a_2,a_3,\ldots$ are digits (i.e. $a_i\in\{0,1,2\ldots,9\}$) in decimal representation of $x$.

Not only that infinite decimal representations can represent finite values, most of the finite values cannot be expressed finitely$^{[4]}$ - irrational numbers. When I say most, what I mean is that if you pick random point on number line, chances are $0$ that you picked rational number. Even rational numbers don't always have finite decimal representation, for example $\frac 1 3 = 0.33333\ldots$

In conclusion...

Expression $\pi d$ is actually the exact value of circumference of a circle - this is the very definition of $\pi$. We actually can't do any better since $\pi$ is irrational. Expressions like $3.14d$, $\frac{22}{7}d$, etc. are just numerical approximates of the exact value which might be used in practical purposes. Can we ever know the exact value of $\pi$? No. Can we get arbitrarily close? Yes!

---

$[1]$ Provably, for any circle the ratio of its circumference to diameter is constant (i.e. circumference of a circle is proportional to its diameter). We denote this ratio by symbol $\pi$.

$[2]$ Imagine that you are measuring something of length $3$ up to precision of, say, $5$ decimals. What you can be sure of is that you will get $3.00000$, but cannot be sure that what you are actually measuring isn't really $3.00000000001$.

${[3]}$ It isn't.

Answer 7

Regarding finite versus infinite decimal expansions, let me raise an issue to ponder:

1. Humans with 10 digits think that $\frac{1}{3}$ has an infinite expansion, $\frac{1}{5}$ has a finite expansion, and $\frac{1}{7}$ has an infinite expansion.
2. Naxaprathians with 12 digits think that $\frac{1}{3}$ has a finite expansion, $\frac{1}{5}$ has an infinite expansion, and $\frac{1}{7}$ has an infinite expansion.
3. Aldeberanian River Mooses, with 21 digits, think that $\frac{1}{3}$ has a finite expansion, $\frac{1}{5}$ has an infinite expansion, and $\frac{1}{7}$ has a finite expansion.

None of them, however, confuse the potentially infinite length of the expansion of a number with the finite length of its geometric measurement along a straight line or a smoothly curving path.

Answer 8

Holding a wrong tool for a task doesn't imply that doing the task is impossible. Meaning, considering a number system that could fail to represent an entity like pi doesn't imply that pi doesn't describe circle.